BUREAU OF RECLAMATION LIBRARY REPORT 79-11

Snowpack optical properties in the infrared

70 s ~ \ * *=> l,***»r AUb?8 1979 'ZSSSff*

For conversion of SI metric units to U.S./British customary units of measurement consult ASTM Standard E380, Metric Practice Guide, published by the American Society for Testing and Materials, 1916 Race St., Philadelphia, Pa. 19103. BUREAU OF RECLAMATION DENVER U6RARY 92028164 isoEfiim

■ ^ ^ l RREL Report 79-11 Oi

Snowpack optical properties in the infrared

Roger H. Berger

May 1979

Prepared for DIRECTORATE OF MILITARY PROGRAMS OFFICE, CHIEF OF ENGINEERS By UNITED STATES ARMY CORPS OF ENGINEERS COLD REGIONS RESEARCH AND ENGINEERING LABORATORY HANOVER, NEW HAMPSHIRE, U.S.A.

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3 SNOWPACK OPTICAL PROPERTIES IN THE INFRARED

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Roger H. Berger s

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK A R E A & WORK UNIT NUMBERS U .S.% rny Cold Regions Research and Engineering Labefatury™ DA Project 4A762730AT42 Hanover, New Hampshire 03755 ^ Technical Area A1, Work Unit 004

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19. K E Y WORDS (Continue on reverse aide if necesaary and identify by block number) Infrared radiation Optical properties Reflection Snow

2Qv ABSTRACT (C o n tìn u e en reverse atxtm ft naceaeary and. identity by block number) A theory of the optical properties of snow in the 2-20 jjtm region of the infrared has been developed. Using this theory it is possible to predict the absorption and scattering coefficients and the emissivity of snow, as a function of the snow parameters of grain size and density, for densities between 0.17 and 0.4 g/cm3. The absorption and scattering coefficients are linearly related to the density and inversely related to the average grain size. The emissivity is independent of grain size and exhibits only a weak dependence upon density.

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This report was prepared by Roger H. Berger, Research Physicist, of the Physical Sciences Branch, Research Division, U.S. Army Cold Regions Research and Engineering Laboratory. Funding for this project was provided by DA Project 4A762730AT42, Design, Construction and Operations Technoiogy for Cold Regions, Technical area A1, ice and Snow Technology, Work Unit 004, Camouflage and Decoy Techniques in Winter Environments. The author gratefully acknowledges the interest and suggestions of Dr. Y.C. Yen and Dr. S.C. Colbeck, who technically reviewed the manuscript of this report.

ii NOMENCLATURE a particle radius C single particle cross section E energy f fraction of the snowpack surface occupied by ice particles H irradiance K absorption coefficient N number of particles/unit volume n Q factor used to convert the transmission coefficient to the transmissivity and assure the con­ servation of radiant energy at an interface R reflectivity T transmissivity X extinction coefficient 7 radiation phase change produced by transmission across a boundary e emissivity r \ radiation phase change produced by reflection at a boundary 0 angle of incidence or coaltitude X wavelength p density a scattering coefficient f skin depth angle of refraction azimuth angle

Subscripts c cavity, denotes the emissivity associated with the pores on the snowpack surface i ice, incident and imaginary used as p }, 0j and /7j, respectively r real s snow, denotes properties associated with the bulk snowpack ab absorption ext extinction sc scattering tr transmitted

iii SNOWPACK OPTICAL PROPERTIES IN THE INFRARED

Roger H. Berger

INTRODUCTION and transmission coefficients and the emissivity of a snowpack from its mean grain size and density. The interaction of electromagnetic radiation with a As shown in Figure 1, the infrared region of the natural snow cover is of considerable practical interest electromagnetic spectrum extends from about 0.8 to and scientific importance. The remote sensing com­ 1000 jum, but absorption by yarious atmospheric con­ munity requires an improved understanding of the op­ stituents limits most infrared remote sensing to two tical properties of snow in order to correctly interpret atmospheric windows at 3-5 and 8-14 /jm. The theory the imagery of snow-covered regions and to distinguish of the infrared optical properties of snow developed in between snowpacks and cloud covers. Radiant energy this report is valid for wavelengths between 2 and 20 transfer is important in the thermodynamics and meta­ pm, the region most useful for remote sensing. morphosis of the snowpack and is therefore of inter­ est to those who study the mechanical and hydrologi­ cal properties of the snowpack. Therefore, the estab­ THE MODEL lishment o f relationships between the optical proper­ ties o f snow and frequently measured snowpack Radiation incident upon the snowpack surface under­ quantities, such as grain size and density, would be of goes either single or multiple interactions as illustrated considerable utility. A model is proposed in this re­ in Figure 2. These multiple interactions have been port for estimating the infrared absorption, reflection treated by Dunkle and Gler (1955) and Bergen (1970)

o Visible A /J. m cm m

------IO11----- IO2 IO3 IO0 IO1 IO2 IOH IO0 IO1 10° a. The electromagnetic spectrum. ----- r ------1 mr * r~ " i '1 t e r S 1— Î " '2 I I1 'U H F1 v h f X-Rays uv P ; I R ■ Radio *

Expanded Area b. Designations within the infrared region o f the electromagnetic spectrum.

c. A tmospheric transmission in the infrared showing the principal atmos­ pheric constituents causing absorp­ tion. The horizontal arrows indi­ cate the atmospheric "windows” through which most remote sensing in the infrared is done.

Figure 7. The electromagnetic spectrum. Incident applicable to this model two conditions have to be satisfied. First, the particles must be large compared to the wavelength of the radiation. Second, the inter­ granular spacing must be large compared to the average grain dimensions. The first condition is easily satisfied, since for most snows, the mean grain diameter is more than an order of magnitude larger than 20 pm, the maximum wavelength considered here. The second implies that the geometrical approach is valid only for low-density snow. The range of snow densities for which this assumption is valid will be examined more closely in relation to the results of the theory.

RADIANT INTERACTIONS

If we consider a single grain within the snowpack, it emits radiation isotropically and is immersed in the flux of radiation from the surrounding grains. Let us con­ sider this external radiation flux as a plane wave and examine the interaction between it and an ice particle. Figure 2. Interaction between electromagnetic radia­ This incident radiation of intensity H will either be tion and the surface layer o f ice particles in a snow- absorbed or scattered and these processes can be des­ pack. The solid lines represent the incident radiation , cribed in terms of the single particle cross sections Cab the dotted lines represent radiation that is specularly for absorption and for scattering. These cross sec­ reflected either singly or m ultiply , and the bold dashed tions are defined by the ratios of the energy absorbed lines represent refracted and absorbed radiation. or scattered to the incident flux: and Giddings and LaChapelle (1961) as a radiative Cab s f ° r absorption (1a) transfer or diffusion problem in a continuum. If the work of these authors is extended from the visible to and infrared wavelengths, problems arise due to the large absorption coefficient of ice in the infrared. Csc = F sc/H for scattering. (1b) A much simpler approach to this scattering prob­ lem and one which is easily adaptable to the longer in­ By the law of conservation of energy these cross sections frared wavelengths is to look at the problem in terms are related to the extinction cross section: of geometrical optics (Bohren and Barkstrom 1974). A rigorous treatment of the snowpack model proposed ^ext ^ab+^sc* (2) in this report would require Mie scattering theory, but the arbitrary shapes of the actual ice grains within the With the initial assumption of a random distribution of snowpack make this refinement unnecessary. grains spaced relatively far apart, these cross sections may be related to the bulk material absorption, scatter­ ing and extinction coefficients by SNOWPACK DESCRIPTION *s = C*N (3a) The snowpack model proposed is a random distri­ bution of spherical ice grains immersed in a nonabsorb­ o , = C scN (3b) ing medium. Below the plane which defines the sur­ face of the snowpack, the grain distribution is homo­ X s = C eMN . (3c) geneous and isotropic and may be characterized by the mean grain radius a, the density of ice pj and the The assumptions of large grain separation and random snow density p s. Since the density of ice is a slowly distribution are necessary so that the contributions varying function of the temperature, p-, can be con­ from the individual grains may be added without con­ sidered a constant. For geometrical optics to be sidering phase . The number of grains per

2 •a

Figure 3. Ray paths illustrating how the incident radi­ ation H interacts with an ice particle. The single particle absorption cross section is calculated by summing the energies [HT(1 -e~K i), H T R e~K i(1-e~K% ..] which are absorbed along successive ray paths [1 , 2 ] o f length £ within the sphere. The scattering cross sec­ tion is calculated by summing the initially reflected energy H R and the transmitted energies [ H T 2e”/cc, H T 2R(e~*'S!J2,...].

a. Real part.

Figure 4. Wavelength dependence o f the complex refractive index o f bulk ice. (Adapted from the measurements o f Schaaf and Williams 1973).

volume N may be expressed as a function of the snow in Figure 3 and on the cover. The sphere is immersed density and grain size: in air with a refractive index o f 1, and the ice has a complex refractive i n d e x , n v-in r The dependence N = 3ps/47ra3Pj. (4) of the refractive index upon wavelength is illustrated in Figure 4. in the succeeding development this wave­ Therefore the absorption, scattering and extinction length dependence is implicit so that all quantities de­ coefficients of the snow can be expressed as functions pending upon the refractive index should be considered o f the snow density and grain size if the single particle spectral quantities. The fractions o f the incident energy cross sections for these processes can be calculated. reflected and transmitted at each air/ice interface are given by the Fresnel coefficients R and T respectively. The irradiance at a distance s along the first ray path CALCULATION OF CROSS SECTIONS inside the sphere (path 1, Figure 3) is

For calculation o f the cross sections, consider a HHs)=HTe~Ks. (5) plane wave incident upon a spherical ice grain as shown

3 Therefore the total energy adsorbed by the sphere /"u = R^2 e/T?l1 = (frcosQ-cos$)/(ncosQ+ along path 1 of length £ is +COS$) (11c) HUb = H T( 1-e 'K i). (6) = T"^2 e/Tl1 = 2cos0/(a7COS0+cos>$') (lid) Similarly for path 2 (see Fig. 3) where the symbols 1 and || refer respectively to the ori­ /y2ab = (/y re -K8)/?(i-e-KC) (7) entation of the electric vector perpendicular and parallel to the plane of incidence. In these formulae both n and 0 are complex, since they are related by Snell’s law: and the total for all internal transits is

sin0 = /7sin$ (12) o o oo

Wab =^ H )ab = (8) Substituting acos^ u-iu and n"= nx-in-x into*the Fresnel y=i2 j-0 formulae and solving for the reflectivity and transmis­ sivity components yields where the sum is the addition of the effects of each transit. For the total energy absorbed by the sphere Rl = [(cos0-*/)2+i/2]/[(cos0+*v)2+i/2] (13a) from the incident plane wave, this expression is inte­ grated over all angles of incidence #¡1 Tl - 4 cos2 0 /[ ( cos0+£v) 2+ i/ 2 ] 0 3 b )

/•it/2 *>2tt £ab = I I c/i// sin^j cos^j /?!, = (13c) Jo Jo jn2+n?)2 cos26-2[u(n2-nf)+2vnrni] cosq+u2+v2 {n2+n?)2cos29+2[u(n2-nf)+2vnrn]\ cos d+u2+v2 a>HrO-e-K«)Y, [/?e~Kg] j (9)

F|| = 4cOS20/|(/72+A72)2 COS20+2[w(A72-rt2) where 0 is the azimuthal angle. +2vntn^\ cosd+u2+v2}. (13d) The total radiation scattered by the sphere may be expressed similarly: In order to satisfy the law of conservation of energy at i *7r/2 *2ir the boundary of the ice particle, the fraction of the in­ c/0j i d\jj sincos0j cident energy which is reflected and the fraction trans­ mitted must add up to unity. This condition, which 0 Jo must be satisfied by each component, is achieved by the addition of factors Q(| and : o o a2/y[/?+/?e'K8^ ( / ? e " K8)i] (10) Ri +Q J i = 1 and Ru+Q\\Tw = 1 O 4 ) /=o where where the first term is due to the reflection of the in­ cident beam and the other terms are due to the radia­ QL =ulcosd (15a) tion that is transmitted after 1,2, 3 ... transits of the sphere. and The reflectivity and transmissivity are derived from

the Fresnel formulae (Born and Wolf 1965): C?n = [£/(/72-/72)+2w7r/7j]/cOS0. (15b)

r i ~ ^ /2 e/1?l = (COS0-A7COS$)/(COS0+ For circularly polarized light, the polarization com­ ponents may be combined to form the total transmis­ +A7COS0) (11a) sivity and reflectivity:

tL = 7 ^ 2e/Vyl = 2cos0 /(cos0+a7cos'$) (11b) T='MQl Tl +C?||r ||) and R = %RL +«„). (16)

4 0° Combining /7cos0 = u-iv with Snell's law and solving ~ for u and v yield:

u = 10~j^n^-nl-sm2 6

+ V(/72-/72-sin20)2+4/72/72| . (17a)

1/ = 0.707|-{/7^-A7?-sin2(9) 1 . _.,14 + V(/72-A72-sin20)2+4/72/72 . (17b)

It will be noted that neither eq 13 nor 17 contain any dependence upon the integration variable \(/, so that eq 9 and 10 may be integrated. These integrals may then be simplified further by examining the ex­ ponential terms. The absorption coefficient of ice in the 3- to 100-/zm wavelength range is between 100 and 14,000 cm-1 (Irvine and Pollock 1968, Schaaf and Williams 1973), and eq 9 and 10 become

7r/2 r(0 , A7r , /7j) sin0cos0c/0 u (18) Figure 5. The reflected energy distribution / as a function o f the angle o f incidence. The and energy has been averaged over the wave­ length interval for each o f the atmospheric /•W 2 windows. Esc = 27ra2H I R (d ,n r, /7j)sin0cos0£/0. Jo (19) This absorption coefficient is dependent upon the ratio Equation 18 gives the total energy absorbed by a of the snow density to the particle size and wavelength. single ice particle of radius a, immersed in a radiation This wavelength dependence is illustrated in Figure 6 field of incident radiant intensity H. The remaining where the quantity K sa/ps is plotted as a function of energy in the incident field is scattered by reflection wavelength. Within the 3-5- and 8-14-ptm wavelength with the energy distribution strongly peaked in the bands for typical snowpack values for the grain size forward direction as shown in Figure 5. and density, the absorption coefficient is in the range The single particle cross sections for absorption and of 2.6 to 12.7 cm“1. Under the same circumstances scattering can now be found by combining eq 1, 18 the scattering due to reflection has a calculated range and 19. of 0.2 to 0.8 cm“1. In addition to the absorption and reflection due to the refractive index of the particles, the diffraction ef­ RESULTS fects must be included when considering forward scattering. All the energy which is incident upon the As a result of the large absorption coefficient, all particle is removed from the incident wave either by the scattering o f the radiation incident upon the par­ absorption or reflection, giving an effective cross sec­ ticle is due to reflection. Combining eq 1 a, 3a,.4 and tion equal to the projected area o f the sphere, na2. 18 yields the absorption coefficient for the ensemble By Babinet's principle the cross section for light d if­ of ice particles: fracted by a large particle, i.e. 1<< x < 2na/\, is equal to its projected area. Therefore the total extinction cross section is twice the geometrical cross section: K s = 1.5ps/ap] /7r, A7j)sin0COS0c/0. (20) Cext = 2TO2. (21)

5 Figure 6. The normalized absorption coefficient as a function o f wavelength. The arrows indicate the two atmospheric win­ dows, 3-5 and 8-14 pm.

The extinction coefficient for snow is EMISSIVITY CALCULATION

* s = * s + 0 ts (2 2 ) This absorption length is of the order of a few par­ ticle diameters and indicates that no significant amount where the subsctipt ts of the scattering coefficient o of radiation at these wavelengths penetrates beyond denotes total scattering effects of the particle. Follow­ the first few ice particles. The snowpack emissivity ing the notation of Bohren and Barkstrom (1974) an can therefore be found by considering only the radia­ effective extinction coefficient may be defined by tive interaction with these surface layers. To account for all the radiation that is absorbed within the surface X * = K s+or+ot (23) layers, three absorption processes have to be considered These processes are 1 ) absorption at initial incidence where the r and t stand for the reflected and trans­ with the ice particles, 2) absorption after one or more mitted parts of the scattering coefficients. Since in reflections within the snowpack, and 3) cavity absorp­ the derivation of eq 19 from eq 10 it was seen that tion in the pore spaces within the snowpack. These the transmitted radiation was negligible, ot is essen­ processes are expressed by the first two terms, the tially zero. From Babinet’s principle and eq 3c the third and fourth terms, and the last term, respectively, effective extinction coefficient is in the following equation which is derived using eq 18 and 19: X* = 0.75ps/ffPj (24) tt/2 and by writing the scattering and absorption coeffici­ 7"(0,/7r,A7j)sm0COS0c/0 ent in terms of this parameter we have /

ar = 0.013 X * and K s = 0.99X * (25) s i n (r/a) r(0,77r,A7j)sin 0 COS0 dd which shows that 99% of the effective extinction is / due to absorption. Defining an extinction and an absorption depth as the reciprocal of the coefficient, these depths are re­ /?(0,/7r,/7j)sin0COS0ûW spectively 0(p)

r ext = 50 mm and t K $ = 3.8 mm. ^ s irfV /tf) +2 ftr2H P B l /?(0,A7r,A7j)sin0cos0 dO These values are calculated for a typical aged snow- Je'(p) pack with density of 350 kg/m3, mean crystal size of 2 mm and ice density of 917 kg/m3. (26) +ec [ 1 - 7tAT2/3 (a2 +r2 ) ] 1.00

Figure 7. The spectraI emissivity o f snow illustrating the de­ pendence upon the density, ps in g/cm3.

where 22.5° over this range. The two projected areas as func­ tions of the density become /•W2 \tA = 2nazHl H \ [R(d,nr,n)+T{0[ ,nt,n^) f = (/Wps) 2/3 (27) Jo P = tt[ 1 / 2 V 2 (/Wps) 1/3+(/Wps) 2/3l (28) sin0cos0c/0 where M = 3/4jrpj. Substituting eq 27 and 28 into eq and 26, the expression for the emissivity becomes

sin‘^(r/i7) e = ir(yWps)2/3 jl- sin 20(p)+[1/V2-(Afps) 1/3] 2/ 11B = 2w 2H I [R{e}nXin^T{etnr}n^] •'o (Afps) 2/3-sin2p/8|

slnOcosQdO tt/8 T(0fnr,n)s\n0cos0d0 and where f is the fraction of the surface area occu­ K pied by the projection of the surface layer of particles, and P is the fraction of the surface area occupied by the portion of the projection of the subsurface par­ e(p) "1 f ticles which is not obscured by the surface layer of T(0,nr,nl) sin0 cos0 c/0J particles. The two functions 0(p) and 0'(p) define the minimum angle of incidence at which total absorption takes place within the snowpack structure. ec is the +{1-*[1/2V 2 (Aips)1/3+2(/W/>s)2'3]|ec. apparent emissivity of a pore which opens onto the (29) surface of the snowpack. In order to evaluate these integrals it is necessary to make a further assumption about the snowpack. The pore emissivity ec is 0.95, based on the calcula­ This assumption is that the average properties of the tions of Chandos and Chandos (1974). The evaluation snowpack may be approximated by a regular array of o f eq 29 for the 3-14-pm spectral region is shown in particles. Since most of the radiation incident upon Figure 7. It can be seen from this figure that the de­ an individual ice particle is absorbed and the absorp­ pendence of emissivity upon the snow density is very weak. Likewise, the emissivity is not a function of tion length is short, the effects of phase coherence may be neglected. Under these conditions 0(p) is a the mean grain size, a. There have been very few emissivity measurements nearly linear function of density and varies between on snow in the 2-14-pm wavelength range and there 50.33° and 51.36° over the density range of 170 to are no published results known to this author which 481 kg/m3. The second function 0 (p ) is a constant report its spectral emission or reflectance and characterize 7 the snow by grain size and density. Values of the ficient of ice is large, the absorption length within an emissivity appearing in the literature range between ice particle is much smaller than the particle dimen­ 0.81 to 0.99, with the higher value being most com­ sions and therefore the absorption coefficient is inde­ mon, but generally no information is included on the pendent of the particle shape. spectral bandwidth or the snow properties. The most The limitations of the theory due to the assumption complete documentation is furnished by Dunkle and of large interparticle spacing are more difficult to assess. Gier (1955) who measured the emissivity of two The development of the bulk material constants from natural and two artificial snows and obtained values the particle interaction cross sections is dependent upon of 0.82, 0.89, 0.81 and 0.95. The lower values apply the particle spacing being large enough so that phase co­ to the finer-grained snow of each variety. The differ­ herence effects may be neglected. Under these condi­ ence in emissivity between the fine- and coarse-grained tions the contribution of each particle may be added to snow was attributed to either particle size or surface get the scattered field of the ensemble. The random roughness, but the change in density might have also distribution of the particle sizes (Bohren and Barkstrom been the cause since no density measurements were 1974, Hodkinson and Greenleaves 1962) and the ran­ made. These results are not sufficiently detailed for a dom shape and orientation (Bohren and Barkstrom definitive comparison with the theory developed in 1974) of the particles within the snowpack may average this report out the phase coherence effects as the density of the For a definitive test of this theory it is necessary to snow increases. Experiments with pigments (Blevin have the snow samples well documented so that the and Brown 1961) show that the reflectance of an en­ effect of grain size and density variations may be semble is independent of particle concentration over known. The wavelength band of the measurement a wide range and then usually decreases at high con­ must also be limited to the region of validity of the centrations, indicating that the theory developed for theory. This is important because inclusion of radia­ snow may be valid for densities up to 40G kg/m3. tion of wavelengths shorter than 2.5 jum will reduce Therefore the absorption and scattering coefficients the measured emissivity since reflectance measure­ and the emissivity values calculated using the theory ments in this region indicate that the emissivity de­ developed here should be useful for densities between creases rapidly. A series of emissivity measurements 170 and 400 kg/m3. The theory also shows that the which can be compared with this theory is in progress. absorption and scattering coefficients are linearly re­ Preliminary results indicate, in agreement with the lated to the snowpack density and inversely related to theory, that the emissivity does not depend upon the the grain size. The emissivity is independent of grain grain size, but the dependence on density is not clearly size and exhibits only a weak dependence upon the defined. These results will be presented in a subsequent snowpack density. report.

LITERATURE CITED DISCUSSION OF RESULTS Bergen, J.D. (1970) A possible relation between grain size, The theory used to derive the optical constants of a density and light attenuation in natural snow cover. Journal of Glaciology, vol. 9, no. 55, p. 154-156. snowpack contains no adjustable parameters and is Blevin, W.R. and W.J. Brown (1961) Effect of particle separa­ based only upon the refractive index of pure bulk ice tion on the reflectance of semi-infinite diffusers. Journal of and the density and average grain size of the snow- the Optica! Society of America, vol. 51, no. 2, p. 129-134. pack. In order to evaluate the validity of the results Bohren, C.F. and B.R. Barkstrom (1974) Theory of the optical derived above, the implications of the initial assump­ properties of snow. Journal of Geophysical Research, vol. 79, no. 30, p. 4527-4535. tion of spherical grains and large interparticle spacing Born, M. and E. Wolf (1965) Principles of optics. New York: must be examined. . If an arbitrarily shaped particle is compared with a Chandos, R.J. and R.E. Chandos (1974) Radiometric proper­ spherical particle of the same volume, the sphere has a ties of isothermal, diffuse wall cavity sources. Applied smaller surface area. Chylek (1977) has shown that Optics, vol. 13, no. 9, p. 2142-2152. Chylek, P. (1977) Extinction cross sections of arbitrarily the average geometrical cross section of the nonspheri- shaped randomly oriented nonspherical particles. Journal cal particle is always larger than that of the correspond­ of the Optica/ Society of America, vol. 67, no. 10, p. 1348- ing spherical particle. It is uncertain what the net ef­ 1350. fect of this on the emissivity would be, since the last Dunkle, R.V. and J.T. Gier (1955) Snow characteristics pro­ term in eq 29 would be decreased while the other ject: Final progress report. University of California, Insti­ tute of Engineering Research, Berkeley, California, series terms would be increased. Since the absorption coef­ 63, issue 5.

8 Giddings, J.C. and E. LaChapelle (1961) Diffusion theory applied to radiant energy distribution and albedo of snow. journal of Geophysical Research, vol. 66, no. 1, p. 181- 189. Hodkinson, J.R. and J. Greenleaves (1962) Computations of light-scattering and extinction by spheres according to dif­ fraction and geometrical optics, and some comparisons with the Mie theory. Journal o f the Optica! Society of America, vol. 53, no. 5, p. 577-588. Irvine, W.M. and J.B. Pollock (1968) Infrared optical proper­ ties of water and ice spheres. Icarus,voh 8, p.324-360. Schaaf, J.W. and D. Williams (1973) Optical constants of ice in the infrared. Journal of the Optical Society of America, vol. 8, no. 6, p. 726-732* Van de Hulst, H.C. (1957) Light scattering by small particles. New York: John Wiley and Sons. A facsimile catalog card in Library of Congress MARC format is reproduced below.

Berger, Roger H. Snowpack optical properties in the infrared / by Roger H. Berger. Hanover, N.H.: U.S. Cold Regions Re­ search and Engineering Laboratory; Springfield, Va.: available from National Technical Information Service, 1979. iii, 12 p., illus., 27 cm. ( CRREL Report 79-11. ) Prepared for Military Engineering Division, Office, Chief of Engineers by Corps of Engineers, U.S. Army Cold Regions Research and Engineering Laboratory, under DA Project 4A762630AT42. Bibliography: p. 8. 1. Infrared radiation. 2. Optical properties. 3. Reflection. 4. Snow. I. United States. Army. Corps of Engineers. II. Cold Regions Research and Engineer­ ing Laboratory, Hanover, N.H. III. Series: CRREL Re­ port 79-11. U.S. GOVERNMENT PRINTING OFFICE: 1979-601-520/91